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   "cells": [
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "*Sebastian Raschka*  \n",
      "last modified: 03/31/2014"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "<hr>\n",
      "I am really looking forward to your comments and suggestions to improve and extend this tutorial! Just send me a quick note   \n",
      "via Twitter: [@rasbt](https://twitter.com/rasbt)  \n",
      "or Email: [bluewoodtree@gmail.com](mailto:bluewoodtree@gmail.com)\n",
      "<hr>"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### Problem Category\n",
      "- Statistical Pattern Recognition   \n",
      "- Supervised Learning  \n",
      "- Parametric Learning  \n",
      "- Bayes Decision Theory  \n",
      "- Univariate data  \n",
      "- 2-class problem\n",
      "- equal variances\n",
      "- equal priors\n",
      "- Gaussian model (2 parameters)\n",
      "- No Risk function\n",
      "<hr>"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "<p><a name=\"sections\"></a>\n",
      "<br></p>\n",
      "\n",
      "# Sections\n",
      "\n",
      "\n",
      "<p>&#8226; <a href=\"#given\">Given information</a><br>\n",
      "&#8226; <a href=\"#deriving_db\">Deriving the decision boundary</a><br>\n",
      "&#8226; <a href=\"#plotting_db\">Plotting the class conditional densities, posterior probabilities, and decision boundary</a><br>\n",
      "&#8226; <a href=\"#classify_rand\">Classifying some random example data</a><br>\n",
      "&#8226; <a href=\"#emp_err\">Calculating the empirical error rate</a><br>\n",
      "\n",
      "  \n",
      "\n",
      "  \n",
      "  \n",
      "\n",
      "\n",
      "\n",
      "<hr>"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "<p><a name=\"given\"></a>\n",
      "<br></p>\n",
      "\n",
      "## Given information:\n",
      "\n",
      "[<a href=\"#sections\">back to top</a>] <br>\n",
      "\n",
      "\n",
      "####model: continuous univariate normal (Gaussian) model for the class-conditional densities\n",
      "\n",
      "\n",
      "$ p(x | \\omega_j) \\sim N(\\mu|\\sigma^2) $\n",
      "\n",
      "$ p(x | \\omega_j) \\sim \\frac{1}{\\sqrt{2\\pi\\sigma^2}} \\exp{ \\bigg[-\\frac{1}{2}\\bigg( \\frac{x-\\mu}{\\sigma}\\bigg)^2 \\bigg] } $\n",
      "\n",
      "\n",
      "####Prior probabilities:\n",
      "\n",
      "$ P(\\omega_1) = P(\\omega_1) = 0.5 $\n",
      "\n",
      "#### Variances of the sample distributions\n",
      "\n",
      "$ \\sigma_1^2 = \\sigma_2^2 = 1 $\n",
      "\n",
      "#### Means of the sample distributions\n",
      "\n",
      "$ \\mu_1 = 4, \\quad \\mu_2 = 10 $\n",
      "\n",
      "\n",
      "<br>"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "<p><a name=\"deriving_db\"></a>\n",
      "<br></p>\n",
      "\n",
      "## Deriving the decision boundary\n",
      "[<a href=\"#sections\">back to top</a>] <br>\n",
      "### Bayes' Rule:\n",
      "\n",
      "\n",
      "$ P(\\omega_j|x) = \\frac{p(x|\\omega_j) * P(\\omega_j)}{p(x)} $\n"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "###Bayes' Decision Rule:\n",
      "\n",
      "Decide $ \\omega_1 $ if $ P(\\omega_1|x) > P(\\omega_2|x) $ else decide $ \\omega_2 $.\n",
      "<br>\n",
      "\n",
      "\n",
      "\\begin{equation}\n",
      "\\Rightarrow \\frac{p(x|\\omega_1) * P(\\omega_1)}{p(x)} > \\frac{p(x|\\omega_2) * P(\\omega_2)}{p(x)}\n",
      "\\end{equation} \n",
      "\n",
      "We can drop $ p(x) $ since it is just a scale factor.\n",
      "\n",
      "\n",
      "$ \\Rightarrow P(x|\\omega_1) * P(\\omega_1) > p(x|\\omega_2) * P(\\omega_2) $\n",
      "\n",
      "$ \\Rightarrow \\frac{p(x|\\omega_1)}{p(x|\\omega_2)} > \\frac{P(\\omega_2)}{P(\\omega_1)} $\n",
      "\n",
      "$ \\Rightarrow \\frac{p(x|\\omega_1)}{p(x|\\omega_2)} > \\frac{0.5}{0.5} $\n",
      "\n",
      "$ \\Rightarrow \\frac{p(x|\\omega_1)}{p(x|\\omega_2)} > 1 $\n",
      "\n",
      "$ \\Rightarrow \\frac{1}{\\sqrt{2\\pi\\sigma_1^2}} \\exp{ \\bigg[-\\frac{1}{2}\\bigg( \\frac{x-\\mu_1}{\\sigma_1}\\bigg)^2 \\bigg] } > \\frac{1}{\\sqrt{2\\pi\\sigma_2^2}} \\exp{ \\bigg[-\\frac{1}{2}\\bigg( \\frac{x-\\mu_2}{\\sigma_2}\\bigg)^2 \\bigg] } $\n",
      "\n",
      "\n",
      "Since we have equal variances, we can drop the first term completely.\n",
      "\n",
      "\n",
      "\n",
      "\n",
      "$ Rightarrow \\exp{ \\bigg[-\\frac{1}{2}\\bigg( \\frac{x-\\mu_1}{\\sigma_1}\\bigg)^2 \\bigg] } >  \\exp{ \\bigg[-\\frac{1}{2}\\bigg( \\frac{x-\\mu_2}{\\sigma_2}\\bigg)^2 \\bigg] } \\quad\\quad \\bigg| \\;ln, \\quad \\mu_1 = 4, \\quad \\mu_2 = 10, \\quad \\sigma=1 $\n",
      "\n",
      "$ \\Rightarrow -\\frac{1}{2} (x-4)^2  > -\\frac{1}{2} (x-10)^2  \\quad \\bigg| \\; \\times(-2) $\n",
      "\n",
      "$ \\Rightarrow (x-4)^2  < (x-10)^2 $\n",
      "\n",
      "$ \\Rightarrow x^2 - 8x + 16 < x^2 - 20x + 100 $\n",
      "\n",
      "$ \\Rightarrow 12x < 84 $\n",
      "\n",
      "$ \\Rightarrow x < 7 $\n"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "<p><a name=\"plotting_db\"></a>\n",
      "<br></p>\n",
      "\n",
      "## Plotting the class conditional densities, posterior probabilities, and decision boundary\n",
      "\n",
      "[<a href=\"#sections\">back to top</a>] <br>"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "%pylab inline\n",
      "\n",
      "import numpy as np\n",
      "from matplotlib import pyplot as plt\n",
      "\n",
      "def pdf(x, mu, sigma):\n",
      "    \"\"\"\n",
      "    Calculates the normal distribution's probability density \n",
      "    function (PDF).  \n",
      "        \n",
      "    \"\"\"\n",
      "    term1 = 1.0 / ( math.sqrt(2*np.pi) * sigma )\n",
      "    term2 = np.exp( -0.5 * ( (x-mu)/sigma )**2 )\n",
      "    return term1 * term2\n",
      "\n",
      "# generating some sample data\n",
      "x = np.arange(0, 100, 0.05)\n",
      "\n",
      "# probability density functions\n",
      "pdf1 = pdf(x, mu=4, sigma=1)\n",
      "pdf2 = pdf(x, mu=10, sigma=1)\n",
      "\n",
      "# Class conditional densities (likelihoods)\n",
      "plt.plot(x, pdf1)\n",
      "plt.plot(x, pdf2)\n",
      "plt.title('Class conditional densities (likelihoods)')\n",
      "plt.ylabel('p(x)')\n",
      "plt.xlabel('random variable x')\n",
      "plt.legend(['p(x|w_1) ~ N(4,1)', 'p(x|w_2) ~ N(10,1)'], loc='upper right')\n",
      "plt.ylim([0,0.5])\n",
      "plt.xlim([0,20])\n",
      "plt.show()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "Populating the interactive namespace from numpy and matplotlib\n"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
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0f5O+0BDeRUdHy0aKtwdnbQgXL15EWFgYoqKiAADffPMNtLS08PHHH8v2cXFx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\nrkJggg==\n",
       "text": [
        "<matplotlib.figure.Figure at 0x103464390>"
       ]
      }
     ],
     "prompt_number": 1
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def posterior(likelihood, prior):\n",
      "    \"\"\"\n",
      "    Calculates the posterior probability (after Bayes Rule) without\n",
      "    the scale factor p(x) (=evidence).  \n",
      "        \n",
      "    \"\"\"\n",
      "    return likelihood * prior\n",
      "\n",
      "# probability density functions\n",
      "posterior1 = posterior(pdf(x, mu=4, sigma=1), 0.5)\n",
      "posterior2 = posterior(pdf(x, mu=10, sigma=1), 0.5)\n",
      "\n",
      "# Class conditional densities (likelihoods)\n",
      "plt.plot(x, posterior1)\n",
      "plt.plot(x, posterior2)\n",
      "plt.title('Posterior Probabilities w. Decision Boundary')\n",
      "plt.ylabel('P(w)')\n",
      "plt.xlabel('random variable x')\n",
      "plt.legend(['P(w_1|x)', 'p(w_2|X)'], loc='upper right')\n",
      "plt.ylim([0,0.5])\n",
      "plt.xlim([0,20])\n",
      "plt.axvline(7, color='r', alpha=0.8, linestyle=':', linewidth=2)\n",
      "plt.annotate('R1', xy=(4, 0.3), xytext=(4, 0.3))\n",
      "plt.annotate('R2', xy=(10, 0.3), xytext=(10, 0.3))\n",
      "plt.show()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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Hh4cH9u3bJ3CkjNWNE0ITo6enh8TERFy5cgVGRkYIDw+XLlu8eDF27twpYHT1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\ndw8WFhZy10ckabt++lQyZ8G9e8Ddu5J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       "text": [
        "<matplotlib.figure.Figure at 0x105256f60>"
       ]
      }
     ],
     "prompt_number": 4
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "<p><a name=\"classify_rand\"></a>\n",
      "<br></p>\n",
      "\n",
      "## Classifying some random example data\n",
      "\n",
      "[<a href=\"#sections\">back to top</a>] <br>"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "#### Note on generating univariate random data from a Normal Distribution\n",
      "\n",
      "We can generate random samples drawn from a Normal distribution via the `np.random.randn()` function. Its default is a standard Normal distribution with $ \\mu = 0 $ and $ \\sigma^2 = 1 $. In order to draw random data from $ N(\\mu, \\sigma^2) $, we use   \n",
      "`sigma * np.random.randn(...) + mu`"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Parameters\n",
      "mu_1 = 4\n",
      "mu_2 = 10\n",
      "sigma_1_sqr = 1\n",
      "sigma_2_sqr = 1\n",
      "\n",
      "# Generating 10 random samples drawn from a Normal Distribution for class 1 & 2\n",
      "x1_samples = sigma_1_sqr**0.5 * np.random.randn(10) + mu_1\n",
      "x2_samples = sigma_1_sqr**0.5 * np.random.randn(10) + mu_2\n",
      "y = [0 for i in range(10)]\n",
      "\n",
      "# Plotting sample data with a decision boundary\n",
      "\n",
      "plt.scatter(x1_samples, y, marker='o', color='green', s=40, alpha=0.5)\n",
      "plt.scatter(x2_samples, y, marker='^', color='blue', s=40, alpha=0.5)\n",
      "plt.title('Classifying random example data from 2 classes')\n",
      "plt.ylabel('P(x)')\n",
      "plt.xlabel('random variable x')\n",
      "plt.legend(['w_1', 'w_2'], loc='upper right')\n",
      "plt.ylim([-0.1,0.1])\n",
      "plt.xlim([0,20])\n",
      "plt.axvline(7, color='r', alpha=0.8, linestyle=':', linewidth=2)\n",
      "plt.annotate('R1', xy=(4, 0.05), xytext=(4, 0.05))\n",
      "plt.annotate('R2', xy=(10, 0.05), xytext=(10, 0.05))\n",
      "plt.show()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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bADB48GDk5eUhKytLP/9uf5ioPtq1a4f4+HhcvnwZNjY2Vbb13X73SnOrbX8U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\n/zLcgIyGXAAAAABJRU5ErkJggg==\n",
       "text": [
        "<matplotlib.figure.Figure at 0x10522a1d0>"
       ]
      }
     ],
     "prompt_number": 17
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "<p><a name=\"emp_err\"></a>\n",
      "<br></p>\n",
      "\n",
      "## Calculating the empirical error rate\n",
      "\n",
      "[<a href=\"#sections\">back to top</a>] <br>"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "w1_as_w2, w2_as_w1 = 0, 0\n",
      "for x1,x2 in zip(x1_samples, x2_samples):\n",
      "    if x1 >= 7:\n",
      "        w1_as_w2 += 1\n",
      "    if x2 < 7:\n",
      "        w2_as_w1 += 1\n",
      "        \n",
      "emp_err =  (w1_as_w2 + w2_as_w1) / float(len(x1_samples) + len(x2_samples))\n",
      "    \n",
      "print('Empirical Error: {}%'.format(emp_err * 100))\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "Empirical Error: 0.0%\n"
       ]
      }
     ],
     "prompt_number": 24
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [],
     "language": "python",
     "metadata": {},
     "outputs": []
    }
   ],
   "metadata": {}
  }
 ]
}